I am working through an exercise book and I have encountered a problem which I have very little idea of what to do, which involves Banach's fixed-point theorem.
Let $f\in C(\mathbb R^n \times \mathbb R^m, \mathbb R^n).$ Further, there exists an $\alpha \in [0,1)$ and to each $\lambda\in\mathbb R^m$ a $q(\lambda) \in [0,\alpha]$ with $$||f(x,\lambda) - f(y,\lambda)||\leq q(\lambda)||x-y||, \quad x,y\in\mathbb R^n .$$ I see here clearly that, through the fixed-point theorem, $f(\cdot, \lambda) $ gives us exactly one fixed point $x(\lambda)\in \mathbb R^n$ for each $\lambda \in \mathbb R^m$.
I am asked to show that $[\lambda \mapsto x(\lambda)] \in C(\mathbb R^m, \mathbb R^n)$. What does this mean? How do I do this? I suppose one could use that $\mathbb R^n \times \mathbb R^m$ is $\mathbb R^{m+n}$ but I'm not sure.
The notation just means that the function $x:\mathbb{R}^m\to\mathbb{R}^n$ which maps each $\lambda\in\mathbb{R}^m$ to a point $x(\lambda)\in\mathbb{R}^n$ is a continuous function.
To get start on showing continuity, you might notice that for any two $\lambda\in\mathbb{R}^m$ that $f(x(\lambda),\lambda)=x(\lambda)$ because $x(\lambda_i)$ is defined to be the fixed point. You might also notice that the norm $||(y,\lambda)- (z,\gamma))||_{m}=\max(||y-z||,||\lambda-\gamma||)$ induces the standard topology on $\mathbb{R}^{n+m}$.